Find the Value
\[ x-\frac{1}{x}=3+2\sqrt{2} \]
Find:
\[ x^3-\frac{1}{x^3} \]
Solution:
Using identity:
\[ a^3-b^3=(a-b)^3+3ab(a-b) \]
Here,
\[ a=x,\quad b=\frac{1}{x},\quad ab=1 \]
\[ x^3-\frac{1}{x^3} = \left(x-\frac{1}{x}\right)^3 +3\left(x-\frac{1}{x}\right) \]
\[ = (3+2\sqrt{2})^3 +3(3+2\sqrt{2}) \]
\[ = (99+70\sqrt{2}) +(9+6\sqrt{2}) \]
\[ = 108+76\sqrt{2} \]